[Paper Review] An inverse Cartier transform via exponential in positive characteristic

Guitang Lan, Mao Sheng+1 more|arXiv (Cornell University)|May 30, 2012

Algebraic Geometry and Number Theory12 references5 citations

TL;DR

This paper constructs a flat bundle from a nilpotent Higgs bundle of exponent $ n \leq p-1 $ on a smooth variety over a perfect field of odd characteristic $ p $, using an exponential-type construction via Frobenius liftings over $ W_2(k) $. The key result shows that this construction is equivalent to the inverse Cartier transform of Ogus–Vologodsky in the geometric case, via an isomorphism induced by relative Frobenius.

ABSTRACT

Let $k$ be a perfect field of odd characteristic $p$ and $X_0$ a smooth connected algebraic variety over $k$ which is assumed to be $W_2(k)$-liftable. In this short note we associate a de Rham bundle to a nilpotent Higgs bundle over $X_0$ of exponent $n\leq p-1$ via the exponential function. Presumably, the association is equivalent to the inverse Cartier transform of A. Ogus and V. Vologodsky for these Higgs bundles. However this point has not been verified in the note. Instead, we show the equivalence of the association with that of Sheng-Xin-Zuo in the geometric case. The construction relies on the cocycle property of the difference of different Frobenius liftings over $W_2(k)$, which plays the key role in the proof of $E_1$-degeration of the Hodge to de Rham spectral sequence of $X_0$ due to P. Deligne and L. Illusie.

Motivation & Objective

• To construct a flat bundle from a nilpotent Higgs bundle of exponent $ n \leq p-1 $ over a smooth variety $ X_0 $ over a perfect field $ k $ of odd characteristic $ p $.
• To provide an alternative construction of the inverse Cartier transform using differential geometric tools and the exponential function.
• To establish equivalence between this construction and the inverse Cartier transform of Ogus–Vologodsky in the geometric case.
• To verify that the resulting connection is integrable and well-defined under change of local bases, relying on the cocycle property of Frobenius liftings.

Proposed method

• Lift the variety $ X_0 $ to a smooth $ W_2(k) $-scheme $ X_1 $, and use local Frobenius liftings $ F_\alpha $ over affine opens $ U'_\alpha $.
• Define a connection $ \nabla_\alpha = d + \frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha) $ on the pullback bundle $ H_\alpha = F_0^*E|_{U_\alpha} $, using the Higgs field $ \theta_\alpha $.
• Prove the connection is well-defined under change of local basis by verifying the transformation law using $ dF_0^*M = 0 $ and the structure of $ \frac{dF_\alpha}{[p]} $.
• Show integrability of the connection by proving $ d\left(\frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha)\right) = 0 $ and $ \left(\frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha)\right)^2 = 0 $, relying on nilpotency and integrability of $ \theta $.
• Construct a global flat bundle by gluing local connections using the relative Frobenius isomorphism $ \tilde{\Phi}_{\tilde{F}_\alpha} $, which is independent of the choice of lifting.
• Establish an isomorphism between the constructed flat bundle and the inverse Cartier transform of Ogus–Vologodsky by showing agreement of transition functions and connections via the exponential series $ \sum \frac{(h_{\alpha\beta}(F_0^*\theta))^i}{i!} $.

Experimental results

Research questions

• RQ1 Can a flat bundle be constructed from a nilpotent Higgs bundle of exponent $ n \leq p-1 $ in positive characteristic using an exponential-type map?
• RQ2 Is this construction equivalent to the inverse Cartier transform of Ogus–Vologodsky in the geometric case?
• RQ3 How does the cocycle property of Frobenius liftings over $ W_2(k) $ ensure the well-definedness and integrability of the constructed connection?
• RQ4 What is the role of the relative Frobenius in inducing an isomorphism between the constructed flat bundle and the inverse Cartier transform?
• RQ5 Does the connection defined via $ \nabla_\alpha = d + \frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha) $ yield a flat structure under the nilpotency and integrability conditions of the Higgs field?

Key findings

• The constructed connection $ \nabla_\alpha $ on $ H_\alpha = F_0^*E|_{U_\alpha} $ is well-defined under change of local basis, as the transformation law holds due to $ dF_0^*M = 0 $.
• The connection $ \nabla_\alpha $ is integrable because $ d\left(\frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha)\right) = 0 $ and $ \left(\frac{dF_\alpha}{[p]}(F_0^*\theta_\alpha)\right)^2 = 0 $, ensured by the nilpotency and integrability of $ \theta $.
• The transition functions of the global flat bundle are given by $ g_{\alpha\beta} = 1 + \sum_{|\underline{j}|=1}^n F_0^*(\theta_\partial^{\underline{j}}) \cdot \frac{z^{\underline{j}}}{\underline{j}\,!} $, matching the exponential series of the difference of Frobenius liftings.
• The constructed flat bundle $ (H_{\text{exp}}, \nabla_{\text{exp}}) $ is naturally isomorphic to the inverse Cartier transform $ (H_{(G,\theta)}, \nabla) $ via the relative Frobenius isomorphism $ \tilde{\Phi}_{F_\alpha} $.
• The connection isomorphism holds because $ \tilde{\Phi}_{F_\alpha}(\nabla_{\text{exp}}(F_0^*e_\alpha)) = \nabla[\tilde{\Phi}_{F_\alpha}(F_0^*e_\alpha]) $, verified by matching the connection formulas.
• The construction relies crucially on the cocycle property of the difference of Frobenius liftings over $ W_2(k) $, which ensures consistency in the gluing process and underlies the $ E_1 $-degeneration of the Hodge-to-de Rham spectral sequence.